{"paper":{"title":"Upper Minkowski dimension estimates for convex restrictions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Zoltan Buczolich","submitted_at":"2016-08-02T15:05:19Z","abstract_excerpt":"We show that there are functions $f$ in the H\\\"older class $C^{ { \\alpha }}[0,1]$, $1< { \\alpha }<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \\subset } [0,1]$ with $ { \\bar { dim }_M } A> { \\alpha }-1$.\n  Our earlier result shows that for the typical/generic $f\\in { C_ { 1 } ^ { { \\alpha } } [0,1] }$, $0\\leq { \\alpha }<2$ there is always a set $A { \\subset } [0,1]$ such that $f|_A$ is convex and $ { \\bar { dim }_M } A=1$.\n  The analogous statement for monotone restrictions is the following: there are functions $f$ in the H\\\"older class $C^{ { \\alpha }}[0,1]$, $1/2 \\leq { \\alp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00858","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}